a) Field of the Invention
This invention relates to a Schwarzschild optical system used in an objective lens system for X-ray microscopes, a demagnifying optical system for X-ray lithographic apparatus and the like.
b) Description of the Prior Art
The Schwarzschild optical system is constructed so that as shown in FIG. 1, a concave mirror 1 having an aperture at its center portion and a convex mirror 2 are coaxially arranged between an object point O and an image point I and in the case where a radiation source is placed at the object point O, light emitted from the radiation source is reflected in order of the concave mirror 1 and the convex mirror 2 and converged at the image point I.
In such a Schwarzschild optical system, conditions for correcting coma and spherical aberrations in a paraxial region are given by A. K. Head (A. K. Head, Proc. Phys. Soc. 70,945 (1957)), which are to satisfy three formulas: EQU .rho.+r+l=.rho..sub.0 +r.sub.0 +l.sub.0 (i) EQU sin .theta.=m sin u (m: magnification) (ii) ##EQU2##
With respect to the reference symbols in these formulas which are as shown in FIG. 1, .rho..sub.0, l.sub.0 and r.sub.0 and .rho., l and r are distances between the object point O and the concave mirror 1, between the concave mirror 1 and the convex mirror 2, and between the convex mirror 2 and the image point I along an optical axis and those along a ray of light connecting the object point O with the image point I, respectively, .theta. and u are angles made by the optical axis with the light ray at the object point O and the image point I, respectively, .theta..sub.1 is the incident angle of the light ray entering the convex mirror 1, and m is the magnification of the Schwarzschild optical system. From these requirements, radii of curvature r.sub.1 and r.sub.2 of the concave mirror 1 and the con mirror 2 are determined by the following formulas (1) and (2): EQU r.sub.1 =2 m l.sub.0 .rho..sub.0 /(m.rho..sub.0 +ml.sub.0 -r.sub.0)(1) EQU r.sub.2 =2 l.sub.0 r.sub.0 /(r.sub.0 +l.sub.0 -m.rho..sub.0)(2)
The Schwarzschild optical system fulfilling the above formulas (1) and (2) has the arrangement referred to as a homocentric type in which the center of curvature of the concave mirror coincides with that of the convex mirror so that the incident angle of the ray incident on each mirror (particularly, the incident angle .theta..sub.1 on the concave mirror 1) is small. A design method of the homocentric type Schwarzschild optical system is proposed by P. Erdos (P. Erdos, Opt. Soc. America 49,877 (1959)). Specifically, when the magnification m is indicated, the radius of curvature r.sub.1 of the concave mirror 1, a distance W from the object point O to a common center of curvature of the concave and convex mirrors, and the distance r.sub.0 from a vertex T of the convex surface of the convex mirror 2 to the image point I are obtained, by ratios with the radius of curvature r.sub.2 of the convex mirror 2, from the following formulas.
When r.sub.2 =1, ##EQU3## EQU W=-r.sub.1 /(2-1/x) (4) EQU r.sub.0 =(1-r.sub.1 .multidot.x)/(2r.sub.1 .multidot.x-1) (5)
Here, ##EQU4##
Tables 1, 2, 3 and 4 show examples of the Schwarzschild optical system having the magnifications of 100.times., 50.times., 20.times. and 10.times. obtained according to these formulas (see FIG. 2).
TABLE 1 ______________________________________ m = 100 r1 = 2.67 W = 0.808 r0 = 79.8 ______________________________________
TABLE 2 ______________________________________ m = 50 r1 = 2.71 W = 0.807 r0 = 39.4 ______________________________________
TABLE 3 ______________________________________ m = 20 r1 = 2.87 W = 0.805 r0 = 15.1 ______________________________________
TABLE 4 ______________________________________ m = 10 r1 = 3.18 W = 0.802 r0 = 7.02 ______________________________________
Apart from this, on the other hand, the Schwarzschild optical system in which aberration is favorably corrected by the use of a so-called automatic design program is available. FIG. 3 and Tables 5, 6 and 7 show examples of such an optical system. Any of these examples is a heterocentric optical system in which the center of curvature of the convex mirror is shifted from that of the concave mirror toward the object point. Further, SPIE Vo. 316,316c (1981), as shown in FIG. 4 and Table 8, discloses another heterocentric optical system in which the center of curvature of the convex mirror is shifted form that of the concave mirror toward the image point.
TABLE 5 ______________________________________ m = 100 r1 = 2.67 W1 = 0.810 W2 = 0.811 r0 = 78.2 ______________________________________
TABLE 6 ______________________________________ m = 50 r1 = 2.86 W1 = 0.808 W2 = 0.819 r0 = 36.8 ______________________________________
TABLE 7 ______________________________________ m = 10 r1 = 3.18 W1 = 0.804 W2 = 0.806 r0 = 4.86 ______________________________________
TABLE 8 ______________________________________ m = 20 r1 = 2.58 W1 = 0.810 W2 = 0.792 r0 = 15.8 ______________________________________
Each of the optical systems diminishes the shift between the centers of curvature of the convex and concave mirrors and has properties very similar to the homocentric optical system. According to the discussion of the inventors of this application, if the values of the radii of curvature r.sub.1 and r.sub.2 are within the range of nearly .+-.10% of those given by formulas (1) and (2), namely, EQU 2m l.sub.0 .rho..sub.0 /(m.rho..sub.0 +ml.sub.0 -r.sub.0).times.0.9.ltoreq.r.sub.1 .ltoreq.2m l.sub.0 .rho..sub.0 /(m.rho..sub.0 +ml.sub.0 -r.sub.0).times.1.1 EQU 2l.sub.0 r.sub.0 /(r.sub.0 +l.sub.0 -m.rho..sub.0).times.0.9.ltoreq.r.sub.2 .ltoreq.2l.sub.0 r.sub.0 /(r.sub.0 +l.sub.0 -m.rho..sub.0).times.1.1,
the argument on the homocentric optical system will apply virtually to this case as it is. As such, the homocentric optical system will be variously discussed in the following description.
The function of the Schwarzschild optical system is as follows:
In FIG. 5, the light emanating from the object point O traverses an effective aperture, shown in hatching, of the outer portion of the concave mirror of the Schwarzschild optical system and is first incident on the mirror surface of the concave mirror 1. After this, the light reflected from the mirror surface at a certain reflectance enters the mirror surface of the concave mirror 2. The light reflected from this mirror surface at a certain reflectance is then collected at the image point I.
Also, as a means for bringing about a considerable reflectance when the light in an X-ray wave band is incident on the mirror surface at a minute angle with respect to the normal of the mirror surface, it is known that a multilayer film reflecting mirror is used in which a multilayer film is formed as lamination on a substrate. The multilayer film has dispersion properties relying on an incident angle of light and on an wavelength of incident light.
In some of multilayer films, for example, as shown in FIG. 6, a pair of layers is formed with two kinds of substances a, b, which are laminated on the substrate 3 by the period of constant thickness. In such an instance, respective thicknesses d.sub.1, d.sub.2 of the substances a, b are optimized by Fresnel's recurrence formula (Takeshi Namioka et al., "Designs, Manufactures and Evaluations of Multilayer Film Mirrors for Soft X-Rays", Journal of the Japanese Society of Precision Engineering, 52/11/1986, pp. 1843-1846) so that the reflectance is maximized when light having a wavelength .lambda. is incident on a film surface at a certain angle .theta..sub.0 made with the normal of the film surface.
Regarding curve (a) in FIG. 7, in the multilayer film comprising 100 pairs of layers of Ni--Sc in which the layer thicknesses d.sub.1, d.sub.2 are optimized by Fresnel's recurrence formula so that .theta..sub.0 =0.degree. when the light of the wavelength .lambda.=39.8 .ANG. is incident on the film surface, namely, so that the reflectance is maximized when the light enters perpendicular to the film surface, incident angle distribution of reflectances calculated by the Fresnel's recurrence formula is shown. In the same manner, curves (b), (c) and (d) in FIG. 7 show the incident angle distribution of reflectances relative to the multilayer such as to be maximized in the cases where .theta..sub.0 is 2.8.degree., 7.4.degree. and 10.degree., respectively. According to these data, when the incident angle .theta..sub.0 providing the maximum reflectance is approximately zero, reflectance distribution is regarded as nearly constant one in the vicinity of vertical incidence and the half width of the reflectance distribution trends to decrease as the incident angle .theta..sub.0 increases.
In addition, the multilayer film having a non-periodic structure proposed in Takeshi Namioka et al., "Developments of Light Sources and Optical Systems for Soft X-Ray Lithography", Report of Research by Scientific Research-Aid Fund for the 1985 Fiscal Year (Test Research) (2)), pp. 1-36, 1986 is also available.
As stated above, the Schwarzschild optical system refers to a vertical incidence type optical system in which the light is incident on each mirror surface at a small angle. Consequently, where the optical system is employed in the X-ray wave band, it is necessary to improve the reflectance by coating of the multilayer film on each mirror surface. Because, however, the multilayer film has the dispersion properties relying on the wavelength and the incident angle in the X-ray region, even though the coating of the multilayer film is applied without any consideration for such properties, the reflectance is not improved on each mirror surface, depending on the relationship with the incident angle of light determined by the wavelength of light and by the positional relation between the reflecting mirrors and the object and image points constituting the optical system, and as a result, an image surface will become dark. In the past, no effort has been made to secure the optimum values of parameters of the multilayer film with which each mirror surface is coated, such as, for example, substances, the number of layers and layer thicknesses.